17.8 Applications

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17.8 Applications

We demonstrated in Section 17.4 that the F ourier series expansion permits the application of the phasor techniques used in ac analysis to circuits containing nonsinusoidal periodic excitations. The Fourier series has many other practical applications, particularly in communications and signal processing. Typical applications include spectrum analysis, filtering, rectification, and harmonic distortion. We will consider two of these: spectrum analyzers and filters.

17.8.1 Spectrum Analyzers

The Fourier series provides the spectrum of a signal. As we have seen, the spectrum consists of the amplitudes and phases of the harmonics versus frequency. By providing the spectrum of a signal f(t), the Fourier series helps us identify the pertinent features of the signal. It demon strates which frequencies are playing an important role in the shape of the output and which ones are not. F or example, audible sounds ha ve significant components in the frequency range of 20 Hz to 15 kHz, while visible light signals range from 105 to 106 GHz. Table 17.4 presents some other signals and the frequency ranges of their components. A periodic function is said to be band-limited if its amplitude spectrum contains only a finite number of coefficients An or cn. In this case, the F ourier series becomes

$$f(t) = \sum_{n=-N}^{N} c_n e^{jn\omega_0 t} = a_0 + \sum_{n=1}^{N} A_n \cos(n\omega_0 t + \phi_n)$$

(17.75)

This shows that we need only 2N + 1 terms (namely, a0, A1, A2, …, AN, 1, 2, …, N) to completely specify f(t) if ω0 is kno wn. This leads to the sampling theorem: a band-limited periodic function whose Fourier series contains N harmonics is uniquely specified by its values at 2N + 1 instants in one period.

A spectrum analyzer is an instrument that displays the amplitude of the components of a signal v ersus frequenc y. It sho ws the v arious frequency components (spectral lines) that indicate the amount of energy at each frequency.

It is unlik e an oscilloscope, which displays the entire signal (all components) versus time. An oscilloscope shows the signal in the time domain, while the spectrum analyzer sho ws the signal in the frequenc y domain. There is perhaps no instrument more useful to a circuit analyst than the spectrum analyzer . An analyzer can conduct noise and spuri ous signal analysis, phase checks, electromagnetic interference and filter examinations, vibration measurements, radar measurements, and more. Spectrum analyzers are commercially a vailable in v arious sizes and shapes. Figure 17.42 displays a typical one.

17.8.2 Filters

Filters are an important component of electronics and communications systems. Chapter 14 presented a full discussion on passive and active filters. Here, we investigate how to design filters to select the fundamental component (or any desired harmonic) of the input signal and reject other harmonics. This filtering process cannot be accomplished without the

TABLE 17.4

Frequency ranges of typical signals.

Signal	Frequency Range
Audible sounds	20 Hz to 15 kHz
AM radio	540–1600 kHz
Video signals	dc to 4.2 MHz
(U.S. standards)
VHF television,	54–216 MHz
FM radio
UHF television	470–806 MHz
Cellular telephone	824–891.5 MHz
Microwaves	2.4–300 GHz
Visible light	105 –106 GHz
X-rays	108 –109 GHz
Short-wave radio	3–36 MHz

Fourier series expansion of the input signal. For the purpose of illustration, we will consider tw o cases, a lo w-pass filter and a band-pass filter. In Example 17.6, we already looked at a high-pass RL filter.

The output of a lo w-pass filter depends on the input signal, the transfer function H(ω) of the filter, and the corner or half-po wer fre quency ωc. We recall that ωc = 1∕RC for an RC passive filter. As shown in Fig. 17.43(a), the low-pass filter passes the dc and low-frequency components, while blocking the high-frequency components. By making ω^c sufficiently large (ωc ≫ ω0, e.g., making C small), a large number of the harmonics can be passed. On the other hand, by making ωc sufficiently small (ωc ≪ ω0), we can block out all the ac components and pass only dc, as shown typically in Fig. 17.43(b). (See Fig. 17.2(a) for the Fourier series expansion of the square wave.)

Figure 17.43

(a) Input and output spectra of a low-pass filter, (b) the low-pass filter passes only the dc component when ω^c ≪ ω0.

Similarly, the output of a bandpass filter depends on the input signal, the transfer function of the filter H(ω), its bandwidth B, and its cen ter frequency ωc. As illustrated in Fig. 17.44(a), the filter passes all the harmonics of the input signal within a band of frequencies ( ω1 < ω< ω2) centered around ωc. We have assumed that ω0, 2ω0, and 3ω0 are within that band. If the filter is made highly selective (B ≪ ω0) and ωc = ω0, where ω0 is the fundamental frequency of the input signal, the filter passes only the fundamental component (n = 1) of the input and blocks out all higher harmonics. As shown in Fig. 17.44(b), with a square wave as input, we obtain a sine wave of the same frequency as the output. (Again, refer to Fig. 17.2(a).)

Figure 17.44

x(t)

1

Figure 17.45 For example 17.14.

‒1 0 2 3

1 (a)

(a) Input and output spectra of a bandpass filter, (b) the bandpass filter passes only the fundamental component when B ≪ ω0.

If the sawtooth waveform in Fig. 17.45(a) is applied to an ideal lo w-pass filter with the transfer function shown in Fig. 17.45(b), determine the output.

t

0 ω

10 (b)

1

│H│

Example 17.14

The input signal in Fig. 17.45(a) is the same as the signal in Fig. 17.9. From Practice Prob. 17.2, we know that the Fourier series expansion is

$$x(t) = \frac{1}{2} - \frac{1}{\pi} \sin \omega_0 t - \frac{1}{2\pi} \sin 2\omega_0 t - \frac{1}{3\pi} \sin 3\omega_0 t - \dots$$

In this section, we have used ωc for the center frequency of the bandpass filter instead of ω0 as in Chapter 14, to avoid confusing ω0 with the fundamental frequency of the input signal.

where the period is T = 1 s and the fundamental frequency is ω0 = 2πrad/s. Inasmuch as the corner frequency of the filter is ωc = 10 rad/s, only the dc component and harmonics with nω0 < 10 will be passed. For n = 2, nω0 = 4π = 12.566 rad/s, which is higher than 10 rad/s, meaning that second and higher harmonics will be rejected. Thus, only the dc and fundamental components will be passed. Hence, the output of the filter is

$$y(t) = \frac{1}{2} - \frac{1}{\pi} \sin 2\pi t$$

Figure 17.46 For Practice Prob. 17.14. Rework Example 17.14 if the low-pass filter is replaced by the ideal bandpass filter shown in Fig. 17.46.

Answer:

$$y(t) = -\frac{1}{3\pi} \sin 3\omega_0 t - \frac{1}{4\pi} \sin 4\omega_0 t - \frac{1}{5\pi} \sin 5\omega_0 t$$

.